We are given that the blood platelet counts of a group of women have a bell-shaped distribution with a mean of 247.3 and a standard deviation of 60.7.

Let X = the blood platelet counts of a group of women

The z-score probability distribution for the normal distribution is given by;

Z = $\frac{X-\mu}{\sigma}$ ~ N(0,1)

where, $\mu$ = population mean = 247.3

$\sigma$ = standard deviation = 60.7

Now, according to the empirical rule;

68% of the data values lie within one standard deviation of the mean.

95% of the data values lie within two standard deviations of the mean.

99.7% of the data values lie within three standard deviations of the mean.

Since it is stated that we have to calculate the approximate percentage of women with platelet counts within 3 standard deviations of the mean, or between 65.2 and 429.4, i.e;

z-score for 65.2 = $\frac{X-\mu}{\sigma}$

= $\frac{65.2-247.3}{60.7}$ = -3

z-score for 429.4 = $\frac{X-\mu}{\sigma}$

= $\frac{429.4-247.3}{60.7}$ = 3

So, it means that the approximate percentage of women with platelet counts within 3 standard deviations of the mean is 99.7%.

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2 years ago